For a simple pendulum, a graph is plotted between its kinetic energy $(K.E.)$ and potential energy $(P.E.)$ against its displacement $d$. Which one of the following represents these correctly?

What would be the period of free oscillations of the system shown here if mass $M_1$ is pulled down a little, force constant of the spring is $k$, mass of fixed pulley is negligible and movable, pulley is smooth.

A constant force produces maximum velocity $v$ on the block connected to the spring of force constant $k$ as shown in the figure. When the force constant of spring becomes $4k$, then find the maximum velocity of the block. Assume that initially the spring is in a relaxed state.

Two point masses $m_1$ and $m_2$ are fixed to a light rod hinged at one end. The masses are at distances $l_1$ and $l_2$, respectively, from the hinge. Find the time period of oscillation (small amplitude) of this system in seconds if $m_1=m_2\text{,} l_1=1 \mathrm{m}\text{,} l_2=3 \mathrm{m}$.

A particle of mass $m$ is suspended at the lower end of a thin rod of negligible mass. The upper end of the rod is free to rotate in the plane of the page about a horizontal axis passing through the point $O$. The spring is undeformed when the rod is vertical as shown in figure. If the period of oscillation of the system is $\mathrm{\pi }\sqrt{\frac{L}{n}}$ when it is slightly displaced from its mean position, then find $n$. Take, $k=\frac{9mgL}{l^2}$and $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.

If velocity of a particle moving along a straight line changes sinusoidally with time as shown in the given graph, find the average speed over time interval, $t=0 \text{to} t=2(2n–1) \mathrm{s}$, $n$ being any positive integer.